Zero Pearson coefficient for strongly correlated growing trees

 

SERGEY DOROGOVTSEV

UNIVERSIDADE DE AVEIRO, PORTUGAL

 

We discuss the Pearson coefficient for complex networks with strong degree--degree correlations, review the Pearson coefficient values, and show that in many important cases, this coefficient turnes out to be zero despite the presence of correlations between nearest neighbor degrees. We obtained Pearson's coefficient of strongly correlated recursive networks growing by preferential attachment of every new vertex by $m$ edges. We found that the Pearson coefficient is exactly zero in the infinite network limit for the recursive trees ($m=1$). If the number of connections of new vertices exceeds one ($m>1$), then the Pearson coefficient in the infinite networks equals zero only when the degree distribution exponent $\gamma$ does not exceed $4$. We calculated the Pearson coefficient for finite networks and observed a slow, power-law like approach to an infinite network limit. Our findings indicate that Pearson's coefficient strongly depends on size and details of networks, which makes this characteristic virtually useless for quantitative comparison of different networks. [1] S. N. Dorogovtsev, A. L. Ferreira, A. V. Goltsev, and J. F. F. Mendes, arXiv:0911.4285. [2] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. Phys. 80, 1275 (2008). [3] S. N. Dorogovtsev, Lectures on Complex Networks (Oxford University Press, Oxford, 2010).